1. ECONOMIC ENVIRONMENTS

We have a society whose members’ welfare is described by a function v(x, e, j), where x is consumption of a resource, e is effort and j is the person's type. The type summarizes all aspects of the person's social and biological environment that influence his behaviour and realization of the outcome, for which society deems him to be not responsible. Effort is a measure of the chosen actions the individual takes, which influence the outcome in a positive way: thus v is an increasing function of x and e. In particular, v must be distinguished from the subjective utility function that represents a person's preference order, which is normally construed to be decreasing in effort. It is incorrect to say, however, that a person is (fully) responsible for his effort. For the distributions of effort will generally be different in different types, and the characteristics of those distributions are therefore an aspect of the type – a circumstance. An individual is only partially responsible for his effort: to wit, I say he is responsible for his rank on the distribution of effort of his type. Denote the set of types by J = {1, 2, . . ., J}. The fraction of type j individuals in the society is f_j.

Consider, first, the pure resource allocation problem. There is an amount of a good, per capita, to be distributed among the individuals. A feasible allocation is a distribution of this resource to the population as a function of their types and their effort. Denote such an allocation by X. In general the distributions of effort in the types are themselves functions of the allocation of the resource: thus, we denote the distribution of effort in type j if the allocation is X by G^j(e; X). The effort expended by an individual of type j who sits at rank y ∈ [0, 1] of the distribution of effort of his type is denoted (G^j)⁻¹(y; X). Because y, not e, is the ethically relevant quantity in our theory, we will denote an allocation as

Thus, a feasible allocation satisfies:

(1.1)

Let me reiterate the important point, which develops a statement made in the Introduction, that we measure effort by the rank of a person's effort on the effort distribution of his type, rather than by his absolute effort. I pointed out that Cohen criticized Dworkin for holding persons entirely responsible for their preferences (unless, Dworkin said, these preferences were compulsions or addictions), because preferences are in part determined by one's environment. For example, think of ‘years of school attendance’ as a measure of effort, where the objective of interest is ‘wage earning capacity’. The distribution of effort in a type is a characteristic of the type, not of any individual within it, and as such, if a person is not to be held responsible for his type, he should not be responsible for characteristics of his type's distribution of effort. Now one major reason that a person of disadvantaged circumstances may choose to drop out of secondary school is that that behaviour is frequent in his type – in other words, it is because he belongs to a type with an effort distribution that has a low mean. He should not be held responsible for that low mean, and so we require a measure of effort which sterilizes out the size of the mean. The obvious choice is to measure a person's effort not by the years of school he attended but by the rank of his years of school attended in the distribution of years of school attended of his type. That is the purpose of the formalism above.

To simplify and make concrete the discussion, I will study a special case, which satisfies:

Assumption A

1. A1. The distributions G^j(e) are independent of X.

2. A2. v(x, e, j)=x^ae^bK_j, for some a, b ∈ (0, 1).

Assumption A1 is important and restrictive, for it says that the distributions of effort are not influenced by the state's policy – and, hopefully, that is not true in reality, because we would like state policy to influence effort. However, for the purposes of the present paper, the simplifying assumption is acceptable. Assumption A2 posits a simple form for the interaction of effort and resources.

We may write the welfareFootnote ¹ of an individual at an allocation X = {x^j(.)|j=1, . . ., J; y ∈ [0, 1]} as

(1.2)

Note that there are two conduits through which a person's type affects his outcome: the direct effect, through the third argument of v (i.e. K_j), and the indirect effect, through the second argument of v (i.e. the effect on the distribution of effort). Empirically, as will be noted below in section 7, the indirect effect is often very important, and so for policy purposes, it is important to use a notation which helps us to remember this.

Here are several examples. v measures a worker's wage at age 30, x is the amount of public resource per annum devoted to his education as a child, e is the amount of education he acquires, and j summarizes the effect of his family background on both the years of education he acquires and perhaps, through other channels, on his wage. It is well known that individuals from poorly educated parents in general acquire less education than individuals from highly educated parents. Thus, the family background (j) affects the distribution of effort, as well as having a direct effect on the development of the child.Footnote ² A second example: v is health status, e is lifestyle quality, j is family background and x is medical resources. Here, I believe the model is too simple: I would rather say that circumstances and effort determine education, education and another effort determine lifestyle, and lifestyle and medical resources determine health. Thus, there are two loci at which effort enters.Footnote ³ A third example: ‘individuals’ are countries, v is GDP per capita, j summarizes the level of development and other physical characteristics and endowments of the country, e is the quality of governance in the country's institutions and x is a level of international aid.Footnote ⁴

Note that in these three applications, v is not a utility function that represents an individual's subjective preference order. In economics, we model education as being costly (a disutility) for the individual, and good governance in a country requires costly effort by politicians and bureaucrats. But, by hypothesis, effort here enters positively into the realization of the outcome. In most of the applications that interest me, v measures some outcome that is objective and observable about individuals, and the function v is not a representation of their subjective preferences. This reflects my intention that the theory be useful for policy makers. The Ministry of Health is interested in maximizing some distribution of life expectancy or quality-years of life, not subjective utility (e.g. it will try to discourage smoking, even though individuals may derive subjective pleasure from smoking); the Ministry of Education is interested in producing students who are well-educated, not happy students; and international lending institutions are interested in maximizing GDP per capita in countries, not the subjective utility of the country's politicians.

The standard economic hypothesis is that the choice of effort by the individual reflects the maximization of some subjective preference order. But that assumption is not necessary for the approach I propose: my approach takes as data not preference orders of individuals, but observed distributions of effort and circumstances. I contend that the theory, as I present it, does not require those who wish to use it to accept the rational-choice explanation of effort. Now it is certainly possible to take a more restrictive approach, which derives effort as the result of the maximization of a person's preference order over goods and effort, in which effort is costly. In this case, one would begin not with the objective functions v, but rather utility functions u, in which differential effort was due to a varying cost parameter in utility functions representing the psychic cost of effort expenditure to the individual. Effort would decrease utility in that formulation, and a person would be held responsible for the rank of his idiosyncratic cost parameter in the distribution of that cost parameter in his type. Although I will take this approach in the tax problem that I describe later in this section, I generally eschew it, as a less desirable and more restrictive approach than the one based on the outcome functions v.

This posture also reflects my preference to apply the theory to problems where outcomes are observable, for I believe that in all policy applications, planners will be concerned to deliver equity (here, equal opportunity) with respect to the achievement of a particular objective, which is the concern of their ministry.Footnote ⁵

That having been said, let me now describe a second economic environment, one designed for treating the optimal tax problem. In this case, I will begin, as students of optimal taxation do, with subjective preferences. Let the preferences of an agent over consumption (x) and years of education (s) be given by , some γ ∈ ℜ₊₊. Call γ the agent's degree of ambition. There is a set of types J = {1, 2, . . ., J} and the distribution function of γ in type j is G^j. The frequency of type j in the population is f_j. The wage a person earns is w=βs, some β > 0. The policy space is the set of affine tax policies, (t,b), where a person's after-tax consumption will be (1 − t)w+b. The tax policies are budget-balanced, so the per capita demogrant (or transfer) b is equal to per capita tax revenues.

Facing a tax policy (t,b), the individual chooses a level of education, the solution of

which is s(t, γ)=(1 − t)βγ. It follows that per capita tax revenues, and b, are given by:

(1.3)

where is the average degree of ambition in the population.

We take people to be in part responsible for their degrees of ambition. As indicated, I define the degree of responsibility of an individual as the rank at which his degree of ambition lies, on the distribution of degrees of ambition of his type; that is, it is y where γ=(G^j)⁻¹(y). Therefore we may write the utility of an individual as a function of the policy t and effort as:

(1.4)

where at the end I have taken a ratio-scale transformation of v (that is, I have transformed it by the multiplicative factor 2/β²). Note that, while effort (s) enters negatively into utility, the effort rank (y) enters positively into the indirect utility v. The less ‘costly’ a person views effort (the larger his γ), the higher his eventual welfare v. However, recall that v is not the preference order over income and effort that directs a person's choices: that is summarized by u. A person is not free to choose his cost parameter – that is something that we view him as responsible for, although we do not provide the reason why we do. In this sense, the classical economic approach, of beginning with preferences, is a sort of Procrustean bed into which we can force the theory of equal opportunity, but it is not as natural, so I think, as beginning with the data of observed effort, circumstances and outcomes. Tradition requires, however, that in the optimal tax problem, we take the preference-based approach.

Finally, I remark that, like almost all the literature to date, I here ignore dynamic issues with regard to implementing equality of opportunity. Presumably, if an equal-opportunity policy succeeds, it will change the distribution of types in the next generation. A policy maker should be sensitive to this issue, and therefore be concerned with the long-run consequences of policy. Burak Ünveren and I (2012) have recently studied this problem, but I will not review that work here.

2. A GENERALIZATION OF THE EQUALITY-OF-OPPORTUNITY APPROACH

I next summarize the theory of equal opportunity, as expounded in Roemer (Reference Roemer1998). The ethical premise is that an individual is not responsible for his type, but is responsible for the rank of the distribution of effort at which he sits, within his type. The resource allocation is intended to compensate individuals for their disadvantageous circumstances, while allowing those who expended higher degrees of effort to benefit vis-à-vis others. Let us call the second feature rewards-to-effort.

I define the following function, associated with an allocation X = {x^j(y)|j ∈ {1, . . ., J}, y ∈ [0, 1]}:

(2.3)

The difference between the welfares of individuals at the same value of y, but in different types, is ethically unjustifiable, according to the theory. In choosing an allocation, we desire to eliminate these differences. This is what Fleurbaey (Reference Fleurbaey2008) calls the compensation principle. On the other hand, it is perfectly all right if those who expend higher effort should do better. After Fleurbaey (Reference Fleurbaey2008), this is called the rewards-to-effort principle.

The compensation principle says we should make the function θ as ‘large’ as possible – thus, maximizing the welfare that the individuals who are the worst-off across types at any degree of effort achieve – while rewards-to-effort says that θ should be an increasing (or non-decreasing) function. There is, however, no unique way to make precise the idea that θ should be as large as possible. I proceed as follows. Let be the set of non-negative, weakly increasing functions on the unit interval. Let be an increasing operator on Θ, that is, θ ≥ θ′ ⇒ Γ(θ) > Γ(θ′), where θ ≥ θ′ means that θ dominates θ′ and is strictly greater on a set of positive measure. Then we can write the general equal-opportunity program as:

The point to be chosen is (θ, X). Notice that if Γ is a concave operator on Θ and v is concave in x, then (GEOp) is a concave program, and can be solved with standard methods.Footnote ⁶

In this article, I will often take Γ to be a member of the family:

(2.4)

with the convention that Γ⁰(θ) = ∫log (θ(y))dy.Footnote ⁷ Γ^ρ is a concave operator for all ρ ∈ (−∞, 1]. Notice that if ρ = 1, we can write (GEOp) as:

where we define . Program (EOp) is what I have defined previously (Roemer Reference Roemer1996, Reference Roemer1998) as the equal-opportunity program. The objective of (EOp) is what Fleurbaey (Reference Fleurbaey2008) calls ‘the mean of mins’, and program (EOp) is an instance of what he calls utilitarian reward.

We can summarize (GEOp) verbally. We associate the following value to a feasible allocation X. Compute the distributions of welfare for each type at X. The function (see its definition in eqn. 2.3) gives the welfare of those in type j, at the yth degree of effort, at the allocation X. Graph these functions for each j. The lower envelope of these functions is the function θ(·; X) associated with this allocation. We measure the ‘size’ of θ using the mapping Γ. The program says: choose the feasible allocation X that makes the associated function θ as large as possible. The function θ gives us the welfare of the worst-off individuals at each degree of effort y, across types. Our concern is to maximize the welfare of these worst-off individuals – but since there is a whole interval of them, we need to maximize some function of their welfares. That is the role of Γ.

This presentation shows that it is the function θ, induced by an allocation, which is of normative interest, and the objective of (EOp) emerges as a natural choice, because our aim is to make θ as large as possible. Of course, the family of evaluation functions {Γ^ρ} is a natural one. Decreasing the value of ρ will flatten out the optimal function θ in (GEOp). As ρ approaches negative infinity, the objective of (GEOp) approaches , which would be the ‘extremist’ maximin objective, where individuals are taken to be not responsible for either their effort or their circumstances.Footnote ⁸

The solution of (GEOp) under Assumption A is given by:

Proposition 1. Assume A. Let Γ = Γ^ρ, some −∞ < ρ ≤ 1.

Then the optimal solution to (GEOp) satisfies:

1. a. θ (y)

2. b. x^j(y) = θ(y)^1/a(K_j(G^j)⁻¹(y)^b)^{− 1/a},

where ‘∝’ means ‘is proportional to.’

(Proofs are given in the appendix.)

Because there are no incentive effects on effort induced by the allocation due to assumption A1, it is obvious that at the solution, the (j,y) constraints in (GEOp) are all binding: that is, at every y, welfare is equalized across j. Notice that θ is strictly increasing, assuming that the distributions of effort are strictly increasing functions, since ρa − 1<0 for all values of ρ ∈ (−∞, 1].

We say that:

Definition. Type j strongly dominates type i if K_j > K_i and G^j FOSDFootnote ⁹Gⁱ.

Strong domination is a feature of many applications. Indeed, if the types are completely ordered by strong dominance, we can say that the society is stratified with respect to the kind of welfare measured by v.

Notice that x^j(y) in Proposition 1 is the product of two terms: the first, θ(y), is increasing in y, while the second term, in the special case that the environment is stratified, is decreasing in j, assuming that the most disadvantaged type is type 1. Thus, the first term implements the rewards-to-effort principle, while the second implements the compensation principle.

We now turn to a second problem that is often more appropriate on these environments. It is often, perhaps usually, not desirable to allow allocations X which are predicated on y as well as j. Consider the following example of the allocation of medical resources. There are two social classes, the Rich and the Poor, who suffer from various diseases: the Poor are susceptible to cancer and tuberculosis, while the Rich suffer only from cancer. In all cases, an unhealthy lifestyle increases the probability of the disease. Here, a general allocation policy would prescribe how much to spend on the treatment of disease d in a person of type j whose lifestyle quality was y. But we do not want medical personnel to be compelled to triage patients on the basis of what their lifestyles have been. So the policy should be at most predicated on d and j. In fact, I would argue that, in this case, we want the policy to be predicated only on the disease – not even on the type. That is, ‘horizontal equity’, a desirable feature in this problem, means that anyone with a given disease should be given the same treatment! The equal-opportunity program – which cannot be formulated as a special case of the problem in this section – will allocate medical expenditures to cancer and tuberculosis across the society so as to compensate the Poor. (Thus, relatively more will be spent on TB vis-à-vis cancer, than would be spent with a utilitarian objective.) A person's effort (lifestyle) will be positively related to his expected health outcome, although policy will not explicitly treat people differently as a function of their lifestyles. (For detailed discussion of this example, see Roemer Reference Roemer2007.)

Often it is the case that predicating the policy on effort violates other social values that we hold – including values of privacy. In the health example, we do not want to compromise the relationship between health providers and patients by forcing the former to triage the latter based upon their lifestyles. Therefore, it is important to study the pure allocation problem when the policies can only be predicated on types: that is, a policy is a vector (x ₁, . . ., x_J) such that everyone in type j receives x_j. Philosophers, in criticizing the theory of equal opportunity, sometime make the mistake of assuming that policy must be predicated on effort – so that (in the health case) we would spend fewer medical resources on those with unhealthy lifestyles (see, for instance, Anderson Reference Anderson1999). But this is a fallacy. If we believe that ‘horizontal equity’ is important – and in this case, we do, for at least the reason I have given – then we can define the domain of policies to require it. Thus, if we do not want to discriminate in treatment against those who have expended low effort, we define problem as:

where Γ = Γ^ρ, some ρ ∈ (−∞, 1].

Notice that, in (GEOp2), we restrict the policy to being constant for each type – that is, it requires horizontal equity within a type with respect to different effort levels.

Although the policy space is much simpler in (GEOp2) than in the problem (GEOp), the solution is more complicated to describe. This is because it is generally impossible to achieve the result that the (j,y) constraints in (GEOp2) are all binding. In the solution of the problem (GEOp), the welfare profiles within types, the functions , when graphed in the plane, coincide (for all j). But in the solution of (GEOp2), this is not so: their graphs will cross each other in myriad ways. Therefore the function θ(y), at the solution, will be the lower envelope of the functions , which will coincide with every function for some values of y. (It is easy to see this must be the case. For suppose not, and some function lies entirely above the lower envelope. Then we can increase the value of the program by taking some resource away from type j and distributing it to the other types.) A description of the solution to (GEOp2) must describe the sets in [0,1] for which each function coincides with θ, which in general can be complicated.

We therefore present a result for a special case only.

Proposition 2. Assume A. Assume that there are two types, and type 2 strongly dominates type 1. Further assume that the function is strictly monotone increasing. Then the solution (x ₁, x ₂, y*) of the following three equations characterizes the solution of (GEOp2):

The function θ(·) coincides with on the interval [0, y*] and with on the interval [y*, 1].

It is obvious in this case that the function θ is strictly increasing at the solution. Notice it follows from the second equation in the proposition's statement that x ₂<x ₁. Indeed, we see that the right-hand side of this equation is the product of two terms, each of which is less than one. The first term implements compensation for the direct effect of circumstances, while the second term implements compensation for the indirect effect. Indeed, the second equation merely states that the welfare of the two types must be equal at the rank y* at which the two functions cross; so we might say that the content of the optimization is primarily expressed in the first equation of the three.

Let us note there is an asymmetry in the two principles of compensation and rewards-to-effort. With respect to compensation, we have a definitive rule: at each effort degree y, attempt to maximize the welfare of the type which is worst-off. We have no comparable, definitive instruction for implementing the rewards-to-effort principle. This corresponds to there being many possible choices for the mapping Γ. I do not believe this is a weakness in the theory – rather, it reflects the fact that there is no natural or focal answer to the question, ‘how much reward does effort deserve?’ I use ‘desert’ explicitly, for I believe the equal-opportunity ethic is predicated upon the view that those who expend (costly) effort deserve to be rewarded; more precisely, they deserve to be rewarded for the effort they voluntarily choose which is not determined by their circumstances, and this is measured by their degree of effort in comparison to efforts of others of their type (hence, their rank on the effort distribution of their type).

Fleurbaey expresses a similar point, when he writes:

There is another, independent reason to allow a degree of freedom in the choice of Γ. In policy applications, the typology is always a finite partition of the set of individuals, and effort is usually measured as the residual explanandum of the objective outcome v after type has been accounted for. Now in reality, we are often unsure about what the actual set of circumstances is. Even if we take a political approach, that I have advocated, and say that circumstances are those aspects of the person's environment that the society views as ones it wishes not to hold persons responsible for, the set of circumstances may be too sparse, because society may be ignorant of some causes of behaviour which, were they understood, would have been deemed to be circumstantial. Thus, the variation in welfare outcomes within types that we observe with a given typology may not be properly described as fully due to variations in effort. This is a reason to be flexible on the choice of Γ. As we choose smaller values of ρ, Γ^ρ reduces the rewards to effort, corresponding to a view that there may be more circumstances than we have delineated. So it may be useful to compute solutions to the programs (GEOp) and (GEOp2) for various values to ρ, and take note of the sensitivity of the optimal solution to this parameter.

Indeed, I now believe the equal-opportunity ethic treats the principles ‘compensation for disadvantage’ and ‘rewards to effort’ asymmetrically. With regard to compensation for disadvantage, the theory has an explicit ideal: render the distributions of the outcome identical across types. But with regard to rewarding effort, the ethic is vague, because we have no clear view of how much reward effort deserves. The incompleteness of equality of opportunity as an ethical doctrine takes its formal form, in the present approach, in the availability of many mappings or operators Γ that may be used to pin the policy recommendation down. My current view is that we must apply some additional principle to determine the degree of acceptable inequality due to effort – such as, after Cohen (Reference Cohen2009), the desire to foster community, which is incompatible with large degrees of inequality.

There is, of course, a long history in political philosophy of proposing theories of rewards to effort. Famously, Aristotle proposed ‘proportionality’. But that proposal is poorly defined: obviously, proportionality depends upon the units in which effort and the relevant welfare are measured. I originally (that is, in Roemer Reference Roemer1996, Reference Roemer1998) resolved the problem of rewards-to-effort with a ‘utilitarian’ principle: in case there is just one type, EOp should maximize the average welfare. This is equivalent to choosing the function Γ(θ)=∫¹₀θ(y)dy – in other words, choose ρ = 1. (To repeat, this is how program (EOp) is a special case of program (GEOp).) Fleurbaey's (2008) solution is to say: in the case there is only one type, assign the resource equally to everyone, a principle he calls ‘liberal reward’. I have become, however, more agnostic as to what comprise the right rewards to effort, and prefer to leave the theory open on that account.

In this section, we have computed optimal policies according to the EOp view. Note, however, that we can easily apply the EOp approach to create a complete ordering over feasible allocations, for any choice of Γ. Let X and X′ be two allocations. Compute θ(·; X) and θ(·; X′), where . We say that X≳X′ ⇔ Γ(θ(·; X)) ≥ Γ(θ(·; X′)).

I wish to remark on the measurability properties that are necessary for the approach taken here to make sense. If we use the family of evaluation operators {Γ^ρ}, the function v must be taken to be ratio-scale measurable: that is we assume that v is specified up to a positive multiplicative constant. The solutions to the programs discussed in this section will be invariant with respect to multiplying the function v by a positive constant, but not with respect to more general transformations. I do not view the measurability restriction on welfare, however, to be a limitation of the present approach because, as I said earlier, in policy applications v is virtually always an objective measure which is indeed ratio-scale measurable. Thus life expectancy can be measured in years or months, wages can be measured per annum or per month, and GDP can be measured in dollars or yen.

3. THE REPRESENTATION OF EOP RULES

To re-state, the key function of interest is , where X is the policy domain, defined by:

for any feasible allocation X = {x^j(y)|j ∈ {1, . . ., J}, y ∈ [0, 1]}. An equal-opportunity rule is an increasing operator . The Γ − EOp program is:

I now argue that the representation of an EOp rule as an increasing operator Γ on the domain Θ is, formally, virtually equivalent to the compensation principle. Exactly what does the compensation principle say? After all, it is a somewhat vague statement – that individuals should be compensated for the disadvantages inherent in their circumstances. Let (R, P, I) be an orderingFootnote ¹⁰ of the set of feasible allocations X. I take the compensation principle to be an undefined requirement on R which at least satisfies the following necessary condition:

Axiom. An ordering R on X satisfies the compensation principle only if it satisfies Dom.

Principle Dom.

A. For any two allocations X, X′ such that XPX′ there exists a set of positive measures Y⊆[0, 1], . (Notation: θ^X(·) ≡ θ(·; X).)

B. For any X, X′ ∈ X such that XIX′, either [which means except possibly on a set of measure zero] or there is a set of positive measure Y such that and a set of positive measure Y′ such that .

The motivation for the axiom is seen by supposing it were not true: if the first part of Dom were violated, it would be the case that there exists X, X′ such that XPX’ but for almost all y ∈ [0, 1], . It would be strange to say that this rule satisfies the compensation principle. For clearly for every y ∈ [0, 1], X′ compensates those at effort tranche y at least as well as X does, and there are some y whom it compensates better. On the other hand, suppose part B of Dom were violated. Then R deems two allocations X and X’ to be indifferent, yet (it follows that) either or . If so, why be indifferent between X and X′?

Let Γ be an operator on the set of functions Θ. Γ trivially induces an order on Θ, which we denote R ^Γ, by:

Definition. An operator Γ on Θ represents an order R on X if

The question is: If an order R on X satisfies principle Dom, is it represented by an increasing order Γ on the space of functions Θ* which are induced by the allocations on the economy?

Proposition 3. Let R be an order on Xsatisfying Dom. Let Θ* = {θ^X(·)|X ∈ X}. Then R is represented by an increasing operator Γ on Θ*. Furthermore, if R is a continuous order, then Γ can represented by a continuous increasing operator .

Now consider the optimization problem: Maximize the order R on X. If R satisfies the compensation principle, then the proposition says there is an increasing operator Γ on Θ such that the solution is equivalent to:

We now study the opposite direction. Suppose that we are given an increasing operator, Γ, on Θ, the set of functions derived from an environment from its feasible allocations. Given two allocations X, X′ ∈ X we define the order R by:

R is well-defined, complete, reflexive and transitive. (R is obviously well-defined, since the mapping X↦θ^X is well-defined.)

It is clear that if we begin with an order R on X which is represented by an operator Γ on Θ, then Γ in turn induces the order R, as here described. Thus R and Γ can legitimately be considered dual.

We have:

Theorem 1. Given a dual pair (R, Γ), the order R on Xsatisfies Dom if and only if the operator Γ on Θ is increasing.

Informally, this says that if we are interested in rules satisfying the compensation principle, and we accept the axiom DOM to capture this principle, then we can restrict ourselves to studying the various increasing operators on Θ.Footnote ¹¹ In other words, if one disagrees with my characterization of the equal-opportunity problem as the maximization of some function of the lower envelope θ, then one must object to axiom DOM. As the latest footnote explains, there is at least one possible objection to DOM.

4. THE OPTIMAL TAX PROBLEM

We continue from equation (1.4). Let us now assume that there is one type, number 1, the most disadvantaged type, whose distribution of effort is first-order-stochastic-dominated (FOSD) by the distributions of effort of all other types. As before, we define the lower envelope function by:

(4.1)

where the second equality follows by FOSD. Therefore, the program (GEOp) becomes:

We have: Proposition 4. Let Γ^ρbe the EOp rule. Then the solution of (GEOp) is the tax policy t which solves:

(4.2)

If ρ = 1, the solution is

It is not possible to compute the optimal tax rate analytically for values of ρ other than one. In Figure 1, I display the optimal tax rates as a function of ρ, for the case where , and G ¹ is the uniform distribution on [0, 2γ¹]. As ρ approaches negative infinity, the optimal tax rate approaches one-half, which is the optimal tax rate for the worst off individual in the economy, whose effort is zero.

Figure 1.

The optimal tax rate approaches 0.5 as ρ → −∞

What happens if there is only one type? It is easy to check that if ρ = 1 the optimal tax rate is zero. (Just verify that (t, ρ)=(0, 1) solves equation (4.2) in this case.) This is not a general fact, but is due to the quasi-linearity of utility. However, if ρ<1, the optimal tax rate will be positive. The explanation is – recall – that we are agnostic about the degree of reward due to effort. In the case of one type, all differences in welfare are due to effort plus the action of the competitive market which sets equilibrium wages proportional to skill. Taking an evaluation function Γ^ρ with ρ<1 is tantamount to saying that the laissez-faire solution delivers too much inequality, that it rewards effort too much. Hence, with such a choice of ρ, the optimal tax rate will be positive, implementing some redistribution of income.

There is nothing sacrosanct about the market allocation, even in the case of one type. From an ethical viewpoint, all that can be said for the laissez-faire allocation is that (under the usual Arrow-Debreu assumptions) it is Pareto efficient. Ethically speaking, it is on a par with any other Pareto efficient allocation, and it may well be normatively sub-par with respect to some non-Pareto-efficient allocations, if we believe that redistribution away from laissez-faire is worth at least the efficiency cost. Of course, to be definitive on this question, one requires a theory of just reward to effort, which we do not possess.

As I have been harping on our not possessing a theory of the just rewards to effort, let me remind readers of the solution that many economists offer to the problem: they say that just rewards to effort are determined by the market, in the case that the original endowments are just. And so it follows that rewards to effort are equal to the value of the marginal product of one's effort (in the usual general-equilibrium model, the value marginal produce equals the wage). Indeed, this is the view that Dworkin (Reference Dworkin1981) endorses. But it is a difficult view to justify. Dworkin attempts to do so by saying that ‘. . . people should pay the price of the life they have decided to lead, measured in what others give up in order that they can do so. That was the point of the auction as a device to establish initial equality of resources’ (Dworkin Reference Dworkin1981: 294). But that claim follows for any Pareto efficient resource allocation: that is, at any Pareto efficient allocation of labour and resources there exist ‘supporting prices’ at which ‘each is paying the price of the life she has decided to lead, measured in what others give up in order that she can do so’. This fact is known as the second theorem of welfare economics. The virtue of the market is that (under the usual assumptions of no externalities, complete information, no public goods or bads, etc.) it ‘finds’ one Pareto efficient allocation with decentralized information. But it does not follow from this that the market properly measures how much compensation effort deserves.

5. FLEURBAEY'S APPROACH

A. The pure allocation problem

We assume the same economic environment as in section 1. Suppose that type 1 is strongly dominated by all other types. We study the egalitarian equivalent (EE) allocation rule, introduced in Fleurbaey (Reference Fleurbaey2008: 65).

For any two allocations X = {x^j(y)} and we say that if X is weakly preferred to according to the lexicographic minimum ordering. The wrinkles we must add to the usual definition – since in this case, we are comparing functions, not countable sequences – involve taking measures and derivatives. Thus, the first test in comparing is to ask whether the minimum value X is greater than the minimum value of . If it is, we are done; if the minimum values are the same, compare the measures of the sets on which the minimum value is attained. If they are not the same, we are done. If they are the same, then we have to look at derivatives to see which allocation increases faster. To give a precise definition of leximin for functions requires some care, but in the examples that we study here, the orderings will be obvious.

To compare two allocations X and according to EE, we first construct two fictitious allocations, , defined as follows:

(5.1)

and we say that iff . What is the allocation ? It is the (generally infeasible) allocation which renders each individual just as well off as he is in the actual allocation X, but under the (counterfactual) condition that he were to have been a member of the most disadvantaged type, and expended the same degree of effort as he actually did. In other words, if he is in fact a member of a more advantaged type, his resource under will be larger than it is under X. is a counterfactual allocation which compensates persons for the disadvantage inherent in their circumstances, but continues to reward them (and hold them responsible) for their effort. Consider an individual in an advantaged type, j. Then K_j > K ₁ and (G^j)⁻¹(y)>(G¹)⁻¹(y). Therefore, to achieve the welfare equality of (5.1), we must have .

Thus, the EE-optimal allocation is the X whose shadow allocation leximin-dominates all other shadow allocations. We have:

Proposition 5. Assume A. In the EE-optimal allocation:

An instructive way of comparing the EE solution to the GEOp solution is to compute the function θ associated with the EE solution, and then to compare it with the function θ for the GEOp solution. Let us consider the special case that G^j is uniformly distributed on the interval [0, A_j] so (G^j)⁻¹(y)=yA_j. Then we can compute from Proposition 5 that:

while from Proposition 1 we have that:

It follows that . From this we have:

Corollary

A. For ρ > 0, θ^GEOpis steeper than θ^EE, thus GEOp rewards effort at a greater rate than EE;

B. For ρ < 0, θ^EEis steeper than θ^GEOp, thus EE rewards effort at a greater rate than GEOp;

C. For ρ = 0, θ^EE=θ^GEOpand the policies are identical.

Proof: Follows from observing whether the function y ^{baρ/(1 − aρ)} is increasing, decreasing or constant.   ■

The coincidence of the EE and GEOp allocation rules (for ρ = 1) in part C is an artefact of the special case. In the more general case that we have been studying, EE does not coincide with Γ^ρ for any ρ.

This raises a more general question: Is EE a special case of GEOp for some operator Γ? The answer is, perhaps surprisingly, affirmative.

To demonstrate this, it is costless to work on a more general space of environments than is specified in assumption A1, and so we do so. Let v(x, (G^j)⁻¹(y), j) denote the outcome and define:

this makes sense because, fixing y and the type 1, v is a strictly increasing function of x and so the inverse function exists. The space of allocations is arbitrary. As usual, we define:

We have:

Theorem 2. Define . If the solution of (GEOp) is unique for Γ = Γ^EE, then the associated allocation is the EE allocation.

The operators Γ^ρ were defined on the domain Θ. The operator Γ^EE is defined on the domain Θ × E, where E is the set of environments E = {e|v, {G^j}, {f_j}}. Furthermore, Γ^EE is only weakly increasing as an operator, in contrast to Γ^ρ.

The distinction between leximin and maximin dissolves on the full space of allocations of the resource allocation problem. On other domains, this may not occur: that is, the premise of theorem 2 may be false. To cope with this, instead of the above formulation, we define an order on Θ by θ≳θ′⇔{R(θ(y), y)}leximin dominates {R(θ′(y), y)}.

This order is (strictly) increasing in θ, and we can write the (GEOp) program with the objective of maximizing ≳ on Θ. The solution is the EE allocation.

This is unsurprising. Resorting to the leximin refinement of maximin in the Fleurbaey definition of EE corresponds to not being able to characterize the EE allocation as the maximization of an operator on Θ. But we can still characterize it as the maximization of an order on Θ.

We next calculate the EE solution when we are restricted to the policy space of proposition 2, where allocations are not predicated upon effort.

Proposition 6. Assume A. Assume that G^j(0) = 0 for all j, and that

1. ■ the functions for j≠1 are monotone increasing in y, and

2. ■ the derivatives exist and are positive.

Then on the policy space , the EE optimal solution is given by x_j∝q ⁻¹_jK ^−1/a_j.

Next, let us calculate the EE solution in the optimal affine tax problem. There is ambiguity with regard to how to define the EE solution for the tax problem: Fleurbaey offers at least three alternative definitions (see Proposition 5.2, p. 140).

I will proceed as follows, motivated by the discussion thus far. For any tax policy, we define the lower envelope function:

If I am correct that we should view the EE rule as a special case of (GEOp), then we can observe the following: regardless of the choice of the operator Γ, we must have t ∈ [0, 1/2]. For note that θ(y; t) is a strictly decreasing function of t for t > 1/2. So, unambiguously, to make θ ‘as large as possible’ means to choose t ∈ [0, 1/2].

To compute the EE solution, we must construct a shadow policy where each individual expends the same degree of effort as at the actual policy, and receives the same utility. I propose to take the shadow policy as a tax policy of t=0 plus a transfer. Thus, for any tax policy (t, b(t)), we first compute the ‘shadow allocation’ X = {x^j(y)} satisfying:

The shadow allocation renders every individual as well off as he is at the tax rate t, were he to have been a member of the most disadvantaged type, taken to be type 1, with his own effort y, but with a lump-sum compensation combined with a tax rate of zero. (Note that his utility before the compensation at a tax rate of zero would have been (G ¹)⁻¹(y).) We next choose that tax policy which produces the shadow allocation X that dominates all other such allocations according to the leximin order.Footnote ¹²

Proposition 7. The EE-optimal solution in the affine tax problem is given by

Note the difference/similarity between this policy and the optimal tax policy of proposition 4.

6. THE ‘MIN-OF-MEANS’ RULE

Another popular rule, originally published by Van de gaer (Reference Van de gaer1993), chooses the policy that solves:

(6.1)

Fleurbaey (Reference Fleurbaey2008) calls this the ‘min-of-means’ rule, because it maximizes the minimum mean welfare across types. Note that this rule commutes the ‘min’ operator and the integral operator from program (EOp).

The obvious appeal of the min-of-means rule is that it is a simple way of rendering the distribution functions of the outcome across types as ‘close’ as possible. Furthermore, it has the attraction of often being an easy rule to compute. But unfortunately, this rule does not maximize any operator Γ on the space of functions Θ. Because I believe, as argued, that responsibility-sensitive egalitarianism must comprise some theory of choosing a maximal θ, this is a demerit of the min-of-means rule.

Theorem 3. There is no operator which ‘represents’ the min-of-means rule on the domain of environments specified by assumption A.

Theorem 3 is proved by constructing an environment, with two types, and finding two allocations, call them X ₁ and X ₂, such that the following hold:

This means there can be no increasing operator on Θ whose solution is the allocation recommended by the min-of-means. The details are found in the appendix.

Despite theorem 3, there is a class of environments on which the min-of-means rule and the mean-of-mins rules coincide. This occurs, for example, in the affine-tax problem. The general feature of these environments is that, under all feasible policies, there is a particular type which is worse off than all other types at every degree of effort. It is often the case that on the set of feasible policies, this is true, and so the min-of-means rule may appear to be a good one, although from the deeper standpoint, I believe theorem 3 shows that it is inadequate.

The conditional equality rule (Fleurbaey Reference Fleurbaey2008: 61) is also not representable.Footnote ¹³

7. FURTHER DISCUSSION

I will review three topics:

• • the importance of the effect of circumstances on effort,

• • the liberal reward principle, and

• • the central focus of responsibility-sensitive allocation rules.